Provided by: pdl_2.085-1ubuntu1_amd64 
NAME
PDL::MatrixOps -- Some Useful Matrix Operations
SYNOPSIS
$inv = $x->inv;
$det = $x->det;
($lu,$perm,$par) = $x->lu_decomp;
$y = lu_backsub($lu,$perm,$z); # solve $x x $y = $z
DESCRIPTION
PDL::MatrixOps is PDL's built-in matrix manipulation code. It contains utilities for many common matrix
operations: inversion, determinant finding, eigenvalue/vector finding, singular value decomposition, etc.
PDL::MatrixOps routines are written in a mixture of Perl and C, so that they are reliably present even
when there is no FORTRAN compiler or external library available (e.g. PDL::Slatec or any of the PDL::GSL
family of modules).
Matrix manipulation, particularly with large matrices, is a challenging field and no one algorithm is
suitable in all cases. The utilities here use general-purpose algorithms that work acceptably for many
cases but might not scale well to very large or pathological (near-singular) matrices.
Except as noted, the matrices are PDLs whose 0th dimension ranges over column and whose 1st dimension
ranges over row. The matrices appear correctly when printed.
These routines should work OK with PDL::Matrix objects as well as with normal PDLs.
TIPS ON MATRIX OPERATIONS
Like most computer languages, PDL addresses matrices in (column,row) order in most cases; this
corresponds to (X,Y) coordinates in the matrix itself, counting rightwards and downwards from the upper
left corner. This means that if you print a PDL that contains a matrix, the matrix appears correctly on
the screen, but if you index a matrix element, you use the indices in the reverse order that you would in
a math textbook. If you prefer your matrices indexed in (row, column) order, you can try using the
PDL::Matrix object, which includes an implicit exchange of the first two dimensions but should be
compatible with most of these matrix operations. TIMTOWDTI.)
Matrices, row vectors, and column vectors can be multiplied with the 'x' operator (which is, of course,
broadcastable):
$m3 = $m1 x $m2;
$col_vec2 = $m1 x $col_vec1;
$row_vec2 = $row_vec1 x $m1;
$scalar = $row_vec x $col_vec;
Because of the (column,row) addressing order, 1-D PDLs are treated as _row_ vectors; if you want a
_column_ vector you must add a dummy dimension:
$rowvec = pdl(1,2); # row vector
$colvec = $rowvec->slice('*1'); # 1x2 column vector
$matrix = pdl([[3,4],[6,2]]); # 2x2 matrix
$rowvec2 = $rowvec x $matrix; # right-multiplication by matrix
$colvec = $matrix x $colvec; # left-multiplication by matrix
$m2 = $matrix x $rowvec; # Throws an error
Implicit broadcasting works correctly with most matrix operations, but you must be extra careful that you
understand the dimensionality. In particular, matrix multiplication and other matrix ops need nx1 PDLs
as row vectors and 1xn PDLs as column vectors. In most cases you must explicitly include the trailing
'x1' dimension in order to get the expected results when you broadcast over multiple row vectors.
When broadcasting over matrices, it's very easy to get confused about which dimension goes where. It is
useful to include comments with every expression, explaining what you think each dimension means:
$x = xvals(360)*3.14159/180; # (angle)
$rot = cat(cat(cos($x),sin($x)), # rotmat: (col,row,angle)
cat(-sin($x),cos($x)));
ACKNOWLEDGEMENTS
MatrixOps includes algorithms and pre-existing code from several origins. In particular, "eigens_sym" is
the work of Stephen Moshier, "svd" uses an SVD subroutine written by Bryant Marks, and "eigens" uses a
subset of the Small Scientific Library by Kenneth Geisshirt. They are free software, distributable under
same terms as PDL itself.
NOTES
This is intended as a general-purpose linear algebra package for small-to-mid sized matrices. The
algorithms may not scale well to large matrices (hundreds by hundreds) or to near singular matrices.
If there is something you want that is not here, please add and document it!
FUNCTIONS
identity
Signature: (n; [o]a(n,n))
Return an identity matrix of the specified size. If you hand in a scalar, its value is the size of the
identity matrix; if you hand in a dimensioned PDL, the 0th dimension is the first two dimensions of the
matrix, with higher dimensions preserved.
stretcher
Signature: (a(n); [o]b(n,n))
$mat = stretcher($eigenvalues);
Return a diagonal matrix with the specified diagonal elements
inv
Signature: (a(m,m); sv opt )
$a1 = inv($a, {$opt});
Invert a square matrix.
You feed in an NxN matrix in $a, and get back its inverse (if it exists). The code is inplace-aware, so
you can get back the inverse in $a itself if you want -- though temporary storage is used either way.
You can cache the LU decomposition in an output option variable.
"inv" uses "lu_decomp" by default; that is a numerically stable (pivoting) LU decomposition method.
OPTIONS:
• s
Boolean value indicating whether to complain if the matrix is singular. If this is false, singular
matrices cause inverse to barf. If it is true, then singular matrices cause inverse to return undef.
• lu (I/O)
This value contains a list ref with the LU decomposition, permutation, and parity values for $a. If
you do not mention the key, or if the value is undef, then inverse calls "lu_decomp". If the key
exists with an undef value, then the output of "lu_decomp" is stashed here (unless the matrix is
singular). If the value exists, then it is assumed to hold the LU decomposition.
• det (Output)
If this key exists, then the determinant of $a get stored here, whether or not the matrix is singular.
det
Signature: (a(m,m); sv opt)
$det = det($a,{opt});
Determinant of a square matrix using LU decomposition (for large matrices)
You feed in a square matrix, you get back the determinant. Some options exist that allow you to cache
the LU decomposition of the matrix (note that the LU decomposition is invalid if the determinant is
zero!). The LU decomposition is cacheable, in case you want to re-use it. This method of determinant
finding is more rapid than recursive-descent on large matrices, and if you reuse the LU decomposition
it's essentially free.
OPTIONS:
• lu (I/O)
Provides a cache for the LU decomposition of the matrix. If you provide the key but leave the value
undefined, then the LU decomposition goes in here; if you put an LU decomposition here, it will be
used and the matrix will not be decomposed again.
determinant
Signature: (a(m,m))
$det = determinant($x);
Determinant of a square matrix, using recursive descent (broadcastable).
This is the traditional, robust recursive determinant method taught in most linear algebra courses. It
scales like O(n!) (and hence is pitifully slow for large matrices) but is very robust because no division
is involved (hence no division-by-zero errors for singular matrices). It's also broadcastable, so you
can find the determinants of a large collection of matrices all at once if you want.
Matrices up to 3x3 are handled by direct multiplication; larger matrices are handled by recursive descent
to the 3x3 case.
The LU-decomposition method "det" is faster in isolation for single matrices larger than about 4x4, and
is much faster if you end up reusing the LU decomposition of $a (NOTE: check performance and broadcasting
benchmarks with new code).
eigens_sym
Signature: ([phys]a(m); [o,phys]ev(n,n); [o,phys]e(n))
Eigenvalues and -vectors of a symmetric square matrix. If passed an asymmetric matrix, the routine will
warn and symmetrize it, by taking the average value. That is, it will solve for 0.5*($a+$a->transpose).
It's broadcastable, so if $a is 3x3x100, it's treated as 100 separate 3x3 matrices, and both $ev and $e
get extra dimensions accordingly.
If called in scalar context it hands back only the eigenvalues. Ultimately, it should switch to a faster
algorithm in this case (as discarding the eigenvectors is wasteful).
The algorithm used is due to J. vonNeumann, which was a rediscovery of Jacobi's Method
<http://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm> .
The eigenvectors are returned in COLUMNS of the returned PDL. That makes it slightly easier to access
individual eigenvectors, since the 0th dim of the output PDL runs across the eigenvectors and the 1st dim
runs across their components.
($ev,$e) = eigens_sym $x; # Make eigenvector matrix
$vector = $ev->slice($n); # Select nth eigenvector as a column-vector
$vector = $ev->slice("($n)"); # Select nth eigenvector as a row-vector
($ev, $e) = eigens_sym($x); # e-vects & e-values
$e = eigens_sym($x); # just eigenvalues
eigens_sym ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all
output ndarrays if the flag is set for any of the input ndarrays.
eigens
Signature: ([phys]a(m); [o,phys]ev(l,n,n); [o,phys]e(l,n))
Real eigenvalues and -vectors of a real square matrix.
(See also "eigens_sym", for eigenvalues and -vectors of a real, symmetric, square matrix).
The eigens function will attempt to compute the eigenvalues and eigenvectors of a square matrix with real
components. If the matrix is symmetric, the same underlying code as "eigens_sym" is used. If
asymmetric, the eigenvalues and eigenvectors are computed with algorithms from the sslib library. If any
imaginary components exist in the eigenvalues, the results are currently considered to be invalid, and
such eigenvalues are returned as "NaN"s. This is true for eigenvectors also. That is if there are
imaginary components to any of the values in the eigenvector, the eigenvalue and corresponding
eigenvectors are all set to "NaN". Finally, if there are any repeated eigenvectors, they are replaced
with all "NaN"s.
Use of the eigens function on asymmetric matrices should be considered experimental! For asymmetric
matrices, nearly all observed matrices with real eigenvalues produce incorrect results, due to errors of
the sslib algorithm. If your assymmetric matrix returns all NaNs, do not assume that the values are
complex. Also, problems with memory access is known in this library.
Not all square matrices are diagonalizable. If you feed in a non-diagonalizable matrix, then one or more
of the eigenvectors will be set to NaN, along with the corresponding eigenvalues.
"eigens" is broadcastable, so you can solve 100 eigenproblems by feeding in a 3x3x100 array. Both $ev and
$e get extra dimensions accordingly.
If called in scalar context "eigens" hands back only the eigenvalues. This is somewhat wasteful, as it
calculates the eigenvectors anyway.
The eigenvectors are returned in COLUMNS of the returned PDL (ie the the 0 dimension). That makes it
slightly easier to access individual eigenvectors, since the 0th dim of the output PDL runs across the
eigenvectors and the 1st dim runs across their components.
($ev,$e) = eigens $x; # Make eigenvector matrix
$vector = $ev->slice($n); # Select nth eigenvector as a column-vector
$vector = $ev->slice("($n)"); # Select nth eigenvector as a row-vector
DEVEL NOTES:
For now, there is no distinction between a complex eigenvalue and an invalid eigenvalue, although the
underlying code generates complex numbers. It might be useful to be able to return complex eigenvalues.
($ev, $e) = eigens($x); # e'vects & e'vals
$e = eigens($x); # just eigenvalues
eigens ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output
ndarrays if the flag is set for any of the input ndarrays.
svd
Signature: (a(n,m); [t]w(wsize); [o]u(n,m); [o,phys]z(n); [o]v(n,n))
($u, $s, $v) = svd($x);
Singular value decomposition of a matrix.
"svd" is broadcastable.
Given an m x n matrix $a that has m rows and n columns (m >= n), "svd" computes matrices $u and $v, and a
vector of the singular values $s. Like most implementations, "svd" computes what is commonly referred to
as the "thin SVD" of $a, such that $u is m x n, $v is n x n, and there are <=n singular values. As long
as m >= n, the original matrix can be reconstructed as follows:
($u,$s,$v) = svd($x);
$ess = zeroes($x->dim(0),$x->dim(0));
$ess->slice("$_","$_").=$s->slice("$_") foreach (0..$x->dim(0)-1); #generic diagonal
$a_copy = $u x $ess x $v->transpose;
If m==n, $u and $v can be thought of as rotation matrices that convert from the original matrix's
singular coordinates to final coordinates, and from original coordinates to singular coordinates,
respectively, and $ess is a diagonal scaling matrix.
If n>m, "svd" will barf. This can be avoided by passing in the transpose of $a, and reconstructing the
original matrix like so:
($u,$s,$v) = svd($x->transpose);
$ess = zeroes($x->dim(1),$x->dim(1));
$ess->slice($_,$_).=$s->slice($_) foreach (0..$x->dim(1)-1); #generic diagonal
$x_copy = $v x $ess x $u->transpose;
EXAMPLE
The computing literature has loads of examples of how to use SVD. Here's a trivial example (used in
PDL::Transform::map) of how to make a matrix less, er, singular, without changing the orientation of the
ellipsoid of transformation:
{ my($r1,$s,$r2) = svd $x;
$s++; # fatten all singular values
$r2 *= $s; # implicit broadcasting for cheap mult.
$x .= $r2 x $r1; # a gets r2 x ess x r1
}
svd ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output
ndarrays if the flag is set for any of the input ndarrays.
lu_decomp
Signature: (a(m,m); [o]lu(m,m); [o]perm(m); [o]parity)
LU decompose a matrix, with row permutation
($lu, $perm, $parity) = lu_decomp($x);
$lu = lu_decomp($x, $perm, $par); # $perm and $par are outputs!
lu_decomp($x->inplace,$perm,$par); # Everything in place.
"lu_decomp" returns an LU decomposition of a square matrix, using Crout's method with partial pivoting.
It's ported from Numerical Recipes. The partial pivoting keeps it numerically stable but means a little
more overhead from broadcasting.
"lu_decomp" decomposes the input matrix into matrices L and U such that LU = A, L is a subdiagonal
matrix, and U is a superdiagonal matrix. By convention, the diagonal of L is all 1's.
The single output matrix contains all the variable elements of both the L and U matrices, stacked
together. Because the method uses pivoting (rearranging the lower part of the matrix for better numerical
stability), you have to permute input vectors before applying the L and U matrices. The permutation is
returned either in the second argument or, in list context, as the second element of the list. You need
the permutation for the output to make any sense, so be sure to get it one way or the other.
LU decomposition is the answer to a lot of matrix questions, including inversion and determinant-finding,
and "lu_decomp" is used by "inv".
If you pass in $perm and $parity, they either must be predeclared PDLs of the correct size ($perm is an
n-vector, $parity is a scalar) or scalars.
If the matrix is singular, then the LU decomposition might not be defined; in those cases, "lu_decomp"
silently returns undef. Some singular matrices LU-decompose just fine, and those are handled OK but give
a zero determinant (and hence can't be inverted).
"lu_decomp" uses pivoting, which rearranges the values in the matrix for more numerical stability. This
makes it really good for large and even near-singular matrices. There is a non-pivoting version
"lu_decomp2" available which is from 5 to 60 percent faster for typical problems at the expense of
failing to compute a result in some cases.
Now that the "lu_decomp" is broadcasted, it is the recommended LU decomposition routine. It no longer
falls back to "lu_decomp2".
"lu_decomp" is ported from Numerical Recipes to PDL. It should probably be implemented in C.
lu_decomp2
Signature: (a(m,m); [o]lu(m,m))
LU decompose a matrix, with no row permutation
($lu, $perm, $parity) = lu_decomp2($x);
$lu = lu_decomp2($x,$perm,$parity); # or
$lu = lu_decomp2($x); # $perm and $parity are optional
lu_decomp($x->inplace,$perm,$parity); # or
lu_decomp($x->inplace); # $perm and $parity are optional
"lu_decomp2" works just like "lu_decomp", but it does no pivoting at all. For compatibility with
"lu_decomp", it will give you a permutation list and a parity scalar if you ask for them -- but they are
always trivial.
Because "lu_decomp2" does not pivot, it is numerically unstable -- that means it is less precise than
"lu_decomp", particularly for large or near-singular matrices. There are also specific types of non-
singular matrices that confuse it (e.g. ([0,-1,0],[1,0,0],[0,0,1]), which is a 90 degree rotation matrix
but which confuses "lu_decomp2").
On the other hand, if you want to invert rapidly a few hundred thousand small matrices and don't mind
missing one or two, it could be the ticket. It can be up to 60% faster at the expense of possible
failure of the decomposition for some of the input matrices.
The output is a single matrix that contains the LU decomposition of $a; you can even do it in-place,
thereby destroying $a, if you want. See "lu_decomp" for more information about LU decomposition.
"lu_decomp2" is ported from Numerical Recipes into PDL.
lu_backsub
Signature: (lu(m,m); perm(m); b(m))
Solve A x = B for matrix A, by back substitution into A's LU decomposition.
($lu,$perm,$par) = lu_decomp($A);
$x = lu_backsub($lu,$perm,$par,$A); # or
$x = lu_backsub($lu,$perm,$B); # $par is not required for lu_backsub
lu_backsub($lu,$perm,$B->inplace); # modify $B in-place
$x = lu_backsub(lu_decomp($A),$B); # (ignores parity value from lu_decomp)
# starting from square matrix A and columns matrix B, with mathematically
# correct dimensions
$A = identity(4) + ones(4, 4);
$A->slice('2,0') .= 0; # break symmetry to see if need transpose
$B = sequence(2, 4);
# all these functions take B as rows, interpret as though notional columns
# mathematically confusing but can't change as back-compat and also
# familiar to Fortran users, so just transpose inputs and outputs
# using lu_backsub
($lu,$perm,$par) = lu_decomp($A);
$x = lu_backsub($lu,$perm,$par, $B->transpose)->transpose;
# or with Slatec LINPACK
use PDL::Slatec;
gefa($lu=$A->copy, $ipiv=null, $info=null);
# 1 = do transpose because Fortran's idea of rows vs columns
gesl($lu, $ipiv, $x=$B->transpose->copy, 1);
$x = $x->inplace->transpose;
# or with LAPACK
use PDL::LinearAlgebra::Real;
getrf($lu=$A->copy, $ipiv=null, $info=null);
getrs($lu, 1, $x=$B->transpose->copy, $ipiv, $info=null); # again, need transpose
$x=$x->inplace->transpose;
# or with GSL
use PDL::GSL::LINALG;
LU_decomp(my $lu=$A->copy, my $p=null, my $signum=null);
# $B and $x, first dim is because GSL treats as vector, higher dims broadcast
# so we transpose in and back out
LU_solve($lu, $p, $B->transpose, my $x=null);
$x=$x->inplace->transpose;
# proof of the pudding is in the eating:
print $A x $x;
Given the LU decomposition of a square matrix (from "lu_decomp"), "lu_backsub" does back substitution
into the matrix to solve "a x = b" for given vector "b". It is separated from the "lu_decomp" method so
that you can call the cheap "lu_backsub" multiple times and not have to do the expensive LU decomposition
more than once.
"lu_backsub" acts on single vectors and broadcasts in the usual way, which means that it treats $y as the
transpose of the input. If you want to process a matrix, you must hand in the transpose of the matrix,
and then transpose the output when you get it back. that is because pdls are indexed by (col,row), and
matrices are (row,column) by convention, so a 1-D pdl corresponds to a row vector, not a column vector.
If $lu is dense and you have more than a few points to solve for, it is probably cheaper to find "a^-1"
with "inv", and just multiply "x = a^-1 b".) in fact, "inv" works by calling "lu_backsub" with the
identity matrix.
"lu_backsub" is ported from section 2.3 of Numerical Recipes. It is written in PDL but should probably
be implemented in C.
simq
Signature: ([phys]a(n,n); [phys]b(n); [o,phys]x(n); int [o,phys]ips(n); int flag)
Solution of simultaneous linear equations, "a x = b".
$a is an "n x n" matrix (i.e., a vector of length "n*n"), stored row-wise: that is, "a(i,j) = a[ij]",
where "ij = i*n + j".
While this is the transpose of the normal column-wise storage, this corresponds to normal PDL usage. The
contents of matrix a may be altered (but may be required for subsequent calls with flag = -1).
$y, $x, $ips are vectors of length "n".
Set "flag=0" to solve. Set "flag=-1" to do a new back substitution for different $y vector using the
same a matrix previously reduced when "flag=0" (the $ips vector generated in the previous solution is
also required).
See also "lu_backsub", which does the same thing with a slightly less opaque interface.
simq ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output
ndarrays if the flag is set for any of the input ndarrays.
squaretotri
Signature: (a(n,n); [o]b(m))
Convert a lower-triangular square matrix to triangular vector storage. Ignores upper half of input.
squaretotri does not process bad values. It will set the bad-value flag of all output ndarrays if the
flag is set for any of the input ndarrays.
AUTHOR
Copyright (C) 2002 Craig DeForest (deforest@boulder.swri.edu), R.J.R. Williams (rjrw@ast.leeds.ac.uk),
Karl Glazebrook (kgb@aaoepp.aao.gov.au). There is no warranty. You are allowed to redistribute and/or
modify this work under the same conditions as PDL itself. If this file is separated from the PDL
distribution, then the PDL copyright notice should be included in this file.
perl v5.38.2 2024-04-10 MatrixOps(3pm)